Algebra Books

Book SynopsisThe most ubiquitous, and perhaps the most intriguing, number pattern in mathematics is the Fibonacci sequence. In this simple pattern beginning with two ones, each succeeding number is the sum of the two numbers immediately preceding it (1, 1, 2, 3, 5, 8, 13, 21, ad infinitum). Far from being just a curiosity, this sequence recurs in structures found throughout nature - from the arrangement of whorls on a pinecone to the branches of certain plant stems. All of which is astounding evidence for the deep mathematical basis of the natural world. With admirable clarity, two veteran math educators take us on a fascinating tour of the many ramifications of the Fibonacci numbers. They begin with a brief history of a distinguished Italian discoverer, who, among other accomplishments, was responsible for popularizing the use of Arabic numerals in the West. Turning to botany, the authors demonstrate, through illustrative diagrams, the unbelievable connections between Fibonacci numbers and natural forms (pineapples, sunflowers, and daisies are just a few examples). In art, architecture, the stock market, and other areas of society and culture, they point out numerous examples of the Fibonacci sequence as well as its derivative, the "golden ratio." And of course in mathematics, as the authors amply demonstrate, there are almost boundless applications in probability, number theory, geometry, algebra, and Pascal's triangle, to name a few.Accessible and appealing to even the most math-phobic individual, this fun and enlightening book allows the reader to appreciate the elegance of mathematics and its amazing applications in both natural and cultural settings.Trade Review""This is a wonderful introduction...You may end up amazed and incredulous." - Leon M. Lederman, Nobel Laureate“The mathematics in this book is a delight: surprising, insightful, and comprehensive… the result is by turns rigorous, entertaining, and eye-poppingly speculative.”-New Scientist “…a work that, although aimed at a general audience and presupposing no knowledge of mathematics beyond the high school precalculus level, succeeds in entertaining all audiences…Educators, as well as the mathematically curious, are encouraged to pick up this volume. The discussions of Fibonacci numbers in nature, art, architecture, and music are very thorough…highly recommended.” -Choice“The authors have breathed life into what could be considered a fairly dry subject by demonstrating how commonplace items make use of the Fibonacci numbers…there is a great deal of math involved but taken step at a time, it is not that difficult to understand and this understanding leads to a an even greater appreciation of everything from a flower garden to classical music. Overall, this is an interesting if challenging read for the layperson and a gold mine for the mathematically inclined.”-Monsters and Critics “…the authors have presented a compelling and well-developed book, and one that might well make converts out of some hard-core math phobics…an elegant book that enhances their argument that mathematics is ‘the queen of sciences'.”-Education Update “…delightful…accessible to anyone who enjoys or enjoyed high school mathematics. Mathematics teachers from middle school through college will find this book fun to read and useful in the classroom. The authors consider more properties, relationships, and applications of the Fibonacci numbers than most other sources do…I enjoyed reading this book…a valuable addition to the mathematical literature.”-Mathematics Teacher

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Book SynopsisBasic Linear Algebra is a text for first year students leading from concrete examples to abstract theorems, via tutorial-type exercises. More exercises (of the kind a student may expect in examination papers) are grouped at the end of each section. The book covers the most important basics of any first course on linear algebra, explaining the algebra of matrices with applications to analytic geometry, systems of linear equations, difference equations and complex numbers. Linear equations are treated via Hermite normal forms which provides a successful and concrete explanation of the notion of linear independence. Another important highlight is the connection between linear mappings and matrices leading to the change of basis theorem which opens the door to the notion of similarity. This new and revised edition features additional exercises and coverage of Cramer's rule (omitted from the first edition). However, it is the new, extra chapter on computer assistance that will be of particular interest to readers: this will take the form of a tutorial on the use of the "LinearAlgebra" package in MAPLE 7 and will deal with all the aspects of linear algebra developed within the book.Trade ReviewFrom the reviews: "It embodies a beautiful, concise and precise treatment of the subject, with succint numerical and algebra worked examples at the right points... an excellent textbook which is also eminently suitable for self-study..." Zentralblatt MATH "This is the second edition of a text for first-year students which covers the main themes of linear algebra in a succinct and readable way. … The book is well-written, with a very high standard of proof-reading, and there are full answers to all the exercises … . It should be welcomed as an excellent introductory textbook which could be used either for self-study and to complement a course of lectures." (Gerry Leversha, The Mathematical Gazette, Vol. 87 (509), 2003)Table of ContentsPreface Forward The Algebra of Matrices Some Applications of Matrices Systems of Linear Equations Invertible Matrices Vector Spaces Linear Mappings The Matrix Connection Determinants Eigenvalues and Eigenvectors The Minimum Polynomial Computer Assistance Solutions to the Exercises index

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Book SynopsisThis textbook offers an innovative approach to abstract algebra, based on a unified treatment of similar concepts across different algebraic structures. This makes it possible to express the main ideas of algebra more clearly and to avoid unnecessary repetition.The book consists of two parts: The Language of Algebra and Algebra in Action. The unified approach to different algebraic structures is a primary feature of the first part, which discusses the basic notions of algebra at an elementary level. The second part is mathematically more complex, covering topics such as the Sylow theorems, modules over principal ideal domains, and Galois theory.Intended for an undergraduate course or for self-study, the book is written in a readable, conversational style, is rich in examples, and contains over 700 carefully selected exercises.Trade Review “This book can be also used by incoming graduate students to refresh their knowledge of Algebra before taking graduate courses. I highly recommend this book for a standard undergraduate algebra course, as well as to students interested in independent study.” (Louisa Catalano, MAA Reviews, July 21, 2019)Table of Contents

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Book SynopsisThis book discusses regular powers and symbolic powers of ideals from three perspectives– algebra, combinatorics and geometry – and examines the interactions between them. It invites readers to explore the evolution of the set of associated primes of higher and higher powers of an ideal and explains the evolution of ideals associated with combinatorial objects like graphs or hypergraphs in terms of the original combinatorial objects. It also addresses similar questions concerning our understanding of the Castelnuovo-Mumford regularity of powers of combinatorially defined ideals in terms of the associated combinatorial data. From a more geometric point of view, the book considers how the relations between symbolic and regular powers can be interpreted in geometrical terms. Other topics covered include aspects of Waring type problems, symbolic powers of an ideal and their invariants (e.g., the Waldschmidt constant, the resurgence), and the persistence of associated primes.Trade Review“This is a very interesting monograph providing a fast introduction to different fields of research devoted to modern aspects and develompents of commutative algebra, algebraic geometry, combinatorics, etc.” (Piotr Pokora, zbMATH 1445.13001, 2020)Table of Contents- Part I Associated Primes of Powers of Ideals - Associated Primes of Powers of Ideals. - Associated Primes of Powers of Squarefree Monomial Ideals. - Final Comments and Further Reading. - Part II Regularity of Powers of Ideals. - Regularity of Powers of Ideals and the Combinatorial Framework. - Problems, Questions, and Inductive Techniques. - Examples of the Inductive Techniques. - Final Comments and Further Reading. - Part III The Containment Problem. - The Containment Problem: Background. - The Containment Problem. - The Waldschmidt Constant of Squarefree Monomial Ideals. - Symbolic Defect. - Final Comments and Further Reading. - Part IV Unexpected Hypersurfaces. - Unexpected Hypersurfaces. - Final Comments and Further Reading.

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Book SynopsisThis textbook emphasizes the interplay between algebra and geometry to motivate the study of advanced linear algebra techniques. Matrices and linear transformations are presented as two sides of the same coin, with their connection motivating inquiry throughout the book. Building on a first course in linear algebra, this book offers readers a deeper understanding of abstract structures, matrix decompositions, multilinearity, and tensors. Concepts draw on concrete examples throughout, offering accessible pathways to advanced techniques. Beginning with a study of vector spaces that includes coordinates, isomorphisms, orthogonality, and projections, the book goes on to focus on matrix decompositions. Numerous decompositions are explored, including the Shur, spectral, singular value, and Jordan decompositions. In each case, the author ties the new technique back to familiar ones, to create a coherent set of tools. Tensors and multilinearity complete the book, with a study of the Kronecker product, multilinear transformations, and tensor products. Throughout, “Extra Topic” sections augment the core content with a wide range of ideas and applications, from the QR and Cholesky decompositions, to matrix-valued linear maps and semidefinite programming. Exercises of all levels accompany each section. Advanced Linear and Matrix Algebra offers students of mathematics, data analysis, and beyond the essential tools and concepts needed for further study. The engaging color presentation and frequent marginal notes showcase the author’s visual approach. A first course in proof-based linear algebra is assumed. An ideal preparation can be found in the author’s companion volume, Introduction to Linear and Matrix Algebra.Trade Review“The book is well-organized. The main notions and results are well-presented, followed by a discussion and problems with detailed solutions. There are many helpful notes and examples. At the end of each section, the reader can frequently find several computational, true/false, or proof exercises. … There are several illustrative and colorful figures. For instance, those illustrating the examples and remarks about the Gershgorin disc theorem or about the geometric interpretation of the positive semidefiniteness are really helpful.” (Carlos M. da Fonseca, zbMATH 1471.15001, 2021)Table of ContentsChapter 1: Vector Spaces.- Chapter 2: Matrix Decompositions.- Chapter 3: Tensors and Multilinearity.- Appendix A: Mathematical Preliminaries.- Appendix B: Additional Proofs.- Appendix C: Selected Exercise Solutions.

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Book SynopsisThis volume collects chapters that examine representation theory as connected with affine Lie algebras and their quantum analogues, in celebration of the impact Vyjayanthi Chari has had on this area. The opening chapters are based on mini-courses given at the conference “Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification”, held on the occasion of Chari’s 60th birthday at the Catholic University of America in Washington D.C., June 2018. The chapters that follow present a broad view of the area, featuring surveys, original research, and an overview of Vyjayanthi Chari’s significant contributions. Written by distinguished experts in representation theory, a range of topics are covered, including: String diagrams and categorification Quantum affine algebras and cluster algebras Steinberg groups for Jordan pairs Dynamical quantum determinants and Pfaffians Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification will be an ideal resource for researchers in the fields of representation theory and mathematical physics.Table of ContentsPublications of Vyjayanthi Chari.- Students of Vyjayanthi Chari.- Part I: Courses.- String Diagrams and Categorification.- Quantum Affine Algebras and Cluster Algebras.- Part II: Surveys.- Work of Vyjayanthi Chari.- Steinberg Groups for Jordan Pairs - An Introduction with Open Problems.- On the Hecke-Algebraic Approach for General Linear Groups over a p-adic Field.- Part III: Papers.- Categorical Representations and Classical p-adic Groups.- Formulae of l-Divided Powers in Uq(sl2),II.- Longest Weyl Group Elements in Action.- Dual Kashiwara Functions for the B(∞) Crystal in the Bipartite Case.- Lusztig's t-Analogue of weight multiplicity via Crystals.- Conormal Varieties on the Cominuscule Grassmannian.- Evaluation Modules for Quantum Toroidal gln Algebras.- Dynamical Quantum Determinants and Pfaffians.

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Book SynopsisThis textbook covers topics of undergraduate mathematics in abstract algebra, geometry, topology and analysis with the purpose of connecting the underpinning key ideas. It guides STEM students towards developing knowledge and skills to enrich their scientific education. In doing so it avoids the common mechanical approach to problem-solving based on the repetitive application of dry formulas. The presentation preserves the mathematical rigour throughout and still stays accessible to undergraduates. The didactical focus is threaded through the assortment of subjects and reflects in the book’s structure.Part 1 introduces the mathematical language and its rules together with the basic building blocks. Part 2 discusses the number systems of common practice, while the backgrounds needed to solve equations and inequalities are developed in Part 3. Part 4 breaks down the traditional, outdated barriers between areas, exploring in particular the interplay between algebra and geometry. Two appendices form Part 5: the Greek etymology of frequent terms and a list of mathematicians mentioned in the book. Abundant examples and exercises are disseminated along the text to boost the learning process and allow for independent work.Students will find invaluable material to shepherd them through the first years of an undergraduate course, or to complement previously learnt subject matters. Teachers may pick’n’mix the contents for planning lecture courses or supplementing their classes.Trade Review“The book being reviewed is a collection of what the author considers to be essential material for undergraduates … . it has to be said that many students will find that there is plenty to learn from this well-written book, which would also be a useful reference text had there been a properly compiled index.” (Peter Shiu, The Mathematical Gazette, Vol. 107 (570), November, 2023)Table of ContentsPart I: Basic Objects and Formalisation - Round-up of Elementary Logic.- Naive Set Theory.- Functions.- More Set Theory and Logic.- Boolean Algebras. Part 2: Numbers and Structures - Intuitive Arithmetics.- Real Numbers.- Totally Ordered Spaces.- Part 3: Elementary Real Functions - Real Polynomials.- Real Functions of One Real Variables.- Algebraic Functions.- Elementary Transcendental Functions.- Complex Numbers.- Enumerative Combinatorics.- Part 4: Geometry through Algebra - Vector Spaces.- Orthogonal Operators.- Actions & Representations.- Elementary Plane Geometry.- Metric Spaces.- Part 5: Appendices - Etymologies.- Index of names.- Main figures.- Glossary.- References.

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Book SynopsisTeaching quantum computation and information is notoriously difficult, because it requires covering subjects from various fields of science, organizing these subjects consistently in a unified way despite their tendency to favor their specific languages, and overcoming the subjects’ abstract and theoretical natures, which offer few examples of actual realizations. In this book, we have organized all the subjects required to understand the principles of quantum computation and information processing in a manner suited to physics, mathematics, and engineering courses as early as undergraduate studies.In addition, we provide a supporting package of quantum simulation software from Wolfram Mathematica, specialists in symbolic calculation software. Throughout the book’s main text, demonstrations are provided that use the software package, allowing the students to deepen their understanding of each subject through self-practice. Readers can change the code so as to experiment with their own ideas and contemplate possible applications. The information in this book reflects many years of experience teaching quantum computation and information. The quantum simulation-based demonstrations and the unified organization of the subjects are both time-tested and have received very positive responses from the students who have experienced them.Trade Review“The book provides an extensive bibliography and index. … this volume is well suited for a advanced graduate or first-year PhD course in quantum mechanics, with ample time available for self-study.” (L.-F. Pau, Computing Reviews, January 30, 2023)Table of Contents1 The Postulates of Quantum Mechanics.- 2 Virtual Realization of Quantum Computers.- 3 Quantum Computation: Overview.- 4 Quantum Algorithms: Introduction.- 5 Quantum Information: Introduction.- 6 Quantum Error Correction Codes: Introduction.- Appendix A Linear Algebra.- Appendix B Mathematica Application Q3.- References.

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Book SynopsisThis book provides an organized exposition of the current state of the theory of commutative semigroup cohomology, a theory which was originated by the author and has matured in the past few years. The work contains a fundamental scientific study of questions in the theory. The various approaches to commutative semigroup cohomology are compared. The problems arising from definitions in higher dimensions are addressed. Computational methods are reviewed. The main application is the computation of extensions of commutative semigroups and their classification.Previously the components of the theory were scattered among a number of research articles. This work combines all parts conveniently in one volume. It will be a valuable resource for future students of and researchers in commutative semigroup cohomology and related areas. Table of Contents- 1. The Beginning. - 2. Beck Cohomology. - 3. Symmetric Cohomology. - 4. Calvo-Cegarra Cohomology. - 5. The Third Cohomology Group. - 6. The Overpath Method. - 7. Symmetric Chains. - 8. Inheritance. - 9. Appendixes.

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Book SynopsisThis book develops a new theory in convex geometry, generalizing positive bases and related to Carathéordory’s Theorem by combining convex geometry, the combinatorics of infinite subsets of lattice points, and the arithmetic of transfer Krull monoids (the latter broadly generalizing the ubiquitous class of Krull domains in commutative algebra)This new theory is developed in a self-contained way with the main motivation of its later applications regarding factorization. While factorization into irreducibles, called atoms, generally fails to be unique, there are various measures of how badly this can fail. Among the most important is the elasticity, which measures the ratio between the maximum and minimum number of atoms in any factorization. Having finite elasticity is a key indicator that factorization, while not unique, is not completely wild. Via the developed material in convex geometry, we characterize when finite elasticity holds for any Krull domain with finitely generated class group $G$, with the results extending more generally to transfer Krull monoids. This book is aimed at researchers in the field but is written to also be accessible for graduate students and general mathematicians.Table of Contents- 1. Introduction. - 2. Preliminaries and General Notation. - 3. Asymptotically Filtered Sequences, Encasement and Boundedness. - 4. Elementary Atoms, Positive Bases and Reay Systems. - 5. Oriented Reay Systems. - 6. Virtual Reay Systems. - 7. Finitary Sets. - 8. Factorization Theory.

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Book SynopsisThis book provides a unique tour of university mathematics with the help of Python. Written in the spirit of mathematical exploration and investigation, the book enables students to utilise Python to enrich their understanding of mathematics through: Calculation: performing complex calculations and numerical simulations instantly Visualisation: demonstrating key theorems with graphs, interactive plots and animations Extension: using numerical findings as inspiration for making deeper, more general conjectures. This book is for all learners of mathematics, with the primary audience being mathematics undergraduates who are curious to see how Python can enhance their understanding of core university material. The topics chosen represent a mathematical overview of what students typically study in the first and second years at university, namely analysis, calculus, vector calculus and geometry, differential equations and dynamical systems, linear algebra, abstract algebra and number theory, probability and statistics. As such, it can also serve as a preview of university mathematics for high-school students. The prerequisites for reading the book are a familiarity with standard A-Level mathematics (or equivalent senior high-school curricula) and a willingness to learn programming. For mathematics lecturers and teachers, this book is a useful resource on how Python can be seamlessly incorporated into the mathematics syllabus, assuming only basic knowledge of programming.Table of Contents1 Analysis.- 2 Calculus.- 3 Vector Calculus and Geometry.- 4 Differential Equations and Dynamical Systems.- 5 Linear Algebra.- 6 Abstract Algebra and Number Theory.- 7 Probability.- 8 Statistics.- Appendix A: Python 101.

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Book SynopsisThis textbook provides an undergraduate introduction to Galois theory and its most notable applications. Galois theory was born in the 19th century to study polynomial equations.

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Book SynopsisPreface.- 1 Vectors.- 2 Inner Product, Orthogonality, Norm.- 3 Matrices.- 4. How Matrices Interact with Inner Products and Norms.- 5 Eigenvalues and Singular Values.- 6 Basics of Optimization.- 7 Introduction to Machine Learning and Data.- 8 Principal Component Analysis.- 9 Graph Theory and Graph-based Learning.- 10 Neural Networks and Deep Learning.- 11 Advanced Optimization.- Bibliography.- Index.

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Book SynopsisThis textbook on algebraic topology updates a popular textbook from the golden era of the Moscow school of I. M. Gelfand. The first English translation, done many decades ago, remains very much in demand, although it has been long out-of-print and is difficult to obtain. Therefore, this updated English edition will be much welcomed by the mathematical community. Distinctive features of this book include: a concise but fully rigorous presentation, supplemented by a plethora of illustrations of a high technical and artistic caliber; a huge number of nontrivial examples and computations done in detail; a deeper and broader treatment of topics in comparison to most beginning books on algebraic topology; an extensive, and very concrete, treatment of the machinery of spectral sequences. The second edition contains an entirely new chapter on K-theory and the Riemann-Roch theorem (after Hirzebruch and Grothendieck).Trade Review“This book is a treasure trove for every mathematician who has to deal with classical algebraic topology and homotopy theory on the research level. … Its style is refreshing and informative, and the reader can feel the authors’ joy at sharing their insight into algebraic topology. … will be a useful addition to any mathematical bookshelf.” (Thomas Hüttemann, Mathematical Reviews, March, 2017)“This book covers all the basic material necessary for complete understanding of the fundamentals of algebraic topology … . This increase in the number of topics has made the book more convenient for serious students not only to extend their knowledge but also to gain insight into the interplay between these three subjects. … This book is designed to help students to select the level of learning subjects they want to reach … .” (Haruo Minami, zbMATH 1346.55001, 2016)Table of ContentsIntroduction.- Homotopy.- Homology.- Spectral Sequences of Fibrations.- Cohomology Operations.- The Adams Spectral Sequence.- K-Theory and Other Extraordinary Cohomology Theories.

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Book SynopsisThis book takes the reader on a journey through the world of college mathematics, focusing on some of the most important concepts and results in the theories of polynomials, linear algebra, real analysis, differential equations, coordinate geometry, trigonometry, elementary number theory, combinatorics, and probability. Preliminary material provides an overview of common methods of proof: argument by contradiction, mathematical induction, pigeonhole principle, ordered sets, and invariants. Each chapter systematically presents a single subject within which problems are clustered in each section according to the specific topic. The exposition is driven by nearly 1300 problems and examples chosen from numerous sources from around the world; many original contributions come from the authors. The source, author, and historical background are cited whenever possible. Complete solutions to all problems are given at the end of the book. This second edition includes new sections on quadratic polynomials, curves in the plane, quadratic fields, combinatorics of numbers, and graph theory, and added problems or theoretical expansion of sections on polynomials, matrices, abstract algebra, limits of sequences and functions, derivatives and their applications, Stokes' theorem, analytical geometry, combinatorial geometry, and counting strategies. Using the W.L. Putnam Mathematical Competition for undergraduates as an inspiring symbol to build an appropriate math background for graduate studies in pure or applied mathematics, the reader is eased into transitioning from problem-solving at the high school level to the university and beyond, that is, to mathematical research. This work may be used as a study guide for the Putnam exam, as a text for many different problem-solving courses, and as a source of problems for standard courses in undergraduate mathematics. Putnam and Beyond is organized for independent study by undergraduate and graduate students, as well as teachers and researchers in the physical sciences who wish to expand their mathematical horizons.Table of ContentsPreface to the Second Edition.- Preface to the First Edition.- A Study Guide.- 1. Methods of Proof.- 2. Algebra.- 3. Real Analysis.- 4. Geometry and Trigonometry.- 5. Number Theory.- 6. Combinatorics and Probability.- Solutions.- Index of Notation.- Index.

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Book SynopsisThis book presents four lectures on recent research in commutative algebra and its applications to algebraic geometry. Aimed at researchers and graduate students with an advanced background in algebra, these lectures were given during the Commutative Algebra program held at the Vietnam Institute of Advanced Study in Mathematics in the winter semester 2013 -2014. The first lecture is on Weyl algebras (certain rings of differential operators) and their D-modules, relating non-commutative and commutative algebra to algebraic geometry and analysis in a very appealing way. The second lecture concerns local systems, their homological origin, and applications to the classification of Artinian Gorenstein rings and the computation of their invariants. The third lecture is on the representation type of projective varieties and the classification of arithmetically Cohen -Macaulay bundles and Ulrich bundles. Related topics such as moduli spaces of sheaves, liaison theory, minimal resolutions, and Hilbert schemes of points are also covered. The last lecture addresses a classical problem: how many equations are needed to define an algebraic variety set-theoretically? It systematically covers (and improves) recent results for the case of toric varieties.Table of Contents1. Notes on Weyl Algebras and D-modules.- 2. Inverse Systems of Local Rings.- 3. Lectures on the Representation Type of a Projective Variety.- 4. Simplicial Toric Varieties which are set-theoretic Complete Intersections.

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Book SynopsisThis book offers a comprehensive introduction to the general theory of C*-algebras and von Neumann algebras. Beginning with the basics, the theory is developed through such topics as tensor products, nuclearity and exactness, crossed products, K-theory, and quasidiagonality. The presentation carefully and precisely explains the main features of each part of the theory of operator algebras; most important arguments are at least outlined and many are presented in full detail.Trade ReviewFrom the reviews: "The volume under review is supposed to cover basics on operator algebras … . Blackadar’s book is very well written and pleasant to read. It is especially suited to readers who already know the basics of operator algebras but who need a reference for some result or who wish to have a unified approach to topics treated by them." (Paul Jolissaint, Mathematical Reviews, Issue 2006 k) "This volume is an important and useful contribution to the literature on C*-algebras and von Neumann algebras. … The book is extremely well written. It can be recommended as a reference to graduate students working in operator algebra theory and to other mathematicians who want to bring themselves up-to-date on the subject." (V. M. Manuilov, Zentralblatt MATH, Vol. 1092 (18), 2006)Table of ContentsOperators on Hilbert Space.- C*-Algebras.- Von Neumann Algebras.- Further Structure.- K-Theory and Finiteness.

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Book SynopsisTable of ContentsÜbersicht.- Übungsaufgaben.- Gruppe A. Vorübungen.- Gruppe B. Grundgesetze des Addierens, Multiplizierens und Potenzierens.- Gruppe C. Die drei Grundoperationen.- Gruppe D. Die drei Grundoperationen. Negative Zahlen.- Gruppe E. Verwandlung von Polynomen in Produkte.- Gruppe F. Rechnen mit Brüchen.- Gruppe G. Rechnen mit Brüchen.- Gruppe H. Lineare Gleichungen mit einer Unbekannten.- Gruppe I. Anwendungen von linearen Gleichungen mit einer Unbekannten.- Gruppe K. Systeme von linearen Gleichungen.- Gruppe L. Rechnen mit Quadratwurzeln.- Gruppe M. Quadratische Gleichungen mit einer Unbekannten.- Gruppe N. Anwendungen von quadratischen Gleichungen mit einer Unbekannten.- Gruppe O. Die lineare und die quadratische Funktion im rechtwinkligen Koordinatensystem.- Gruppe P. Darstellung von Funktionen im rechtwinkligen Koordinatensystem.- Gruppe Q. Potenzieren mit ganzzahligen Exponenten.- Gruppe R. Radizieren.- Gruppe S. Potenzieren mit gebrochenen Exponenten.- Gruppe T. Logarithmen.- Gruppe U. Logarithmen. Potenz- und Exponentialfunktion.- Gruppe V. Geometrische Reihen.- Gruppe W. Übungen zur Wiederholung.- Prüfungsaufgaben.- 22 Gruppen (mit Lösungen).- Lösungen zu den Übungsaufgaben.

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Book SynopsisThis book presents the Determinant theory with the use of hypercomplex Grassmann figures. It simplifies the proof of many determinant features. It also states several ways of solutions of linear algebraic systems, most of which are connected with the Determinant theory. The teaching experience in technical universities has demonstrated that the suggested way of presenting the Determinant theory is understood more easily. The title of the book is explained by a desire to attract readers' attention to the well-known in the algebra sphere while introducing a new and more convenient way to present the Determinant theory for technical universities.

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Book SynopsisFourth edition of Topics in Abstract Algebra aligns with CBCS syllabus, includes Automorphism of Groups. Ideal for UG Honours/Advanced level students, competitive exam takers. Features worked-out exercises, MCQs, online resources, historical insights. Market: UG and PG Mathematics.

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Book SynopsisThis is the first volume of the two-volume book on linear algebra, in the University of Tokyo (UTokyo) Engineering Course.The objective of this volume is to present, from the engineering viewpoint, the standard mathematical results in linear algebra such as those on systems of equations and eigenvalue problems. In addition to giving mathematical theorems and formulas, it explains how the mathematical concepts such as rank, eigenvalues, and singular values are linked to engineering applications and numerical computations.In particular, the following four aspects are emphasized.

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Book SynopsisThis is the second volume of the two-volume book on linear algebra in the University of Tokyo (UTokyo) Engineering Course.The objective of this second volume is to branch out from the standard mathematical results presented in the first volume to illustrate useful specific topics pertaining to engineering applications. While linear algebra is primarily concerned with systems of equations and eigenvalue problems for matrices and vectors with real or complex entries, this volumes covers other topics such as matrices and graphs, nonnegative matrices, systems of linear inequalities, integer matrices, polynomial matrices, generalized inverses, and group representation theory.The chapters are, for the most part, independent of each other, and can be read in any order according to the reader's interest. The main objective of this book is to present the mathematical aspects of linear algebraic methods for engineering that will potentially be effective in various application areas.

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Book SynopsisThis book provides a comprehensive knowledge of linear algebra for graduate and undergraduate courses. As a self-contained text, it aims at covering all important areas of the subject, including algebraic structures, matrices and systems of linear equations, vector spaces, linear transformations, dual and inner product spaces, canonical, bilinear, quadratic, sesquilinear, Hermitian forms of operators and tensor products of vector spaces with their algebras. The last three chapters focus on empowering readers to pursue interdisciplinary applications of linear algebra in numerical methods, analytical geometry and in solving linear system of differential equations. A rich collection of examples and exercises are present at the end of each section to enhance the conceptual understanding of readers. Basic knowledge of various notions, such as sets, relations, mappings, etc., has been pre-assumed.Table of Contents1. Algebraic Structures2. Matrices and Systems of Linear Equations3. Vector Spaces4. Linear Transformations5. Dual Spaces6. Inner Product Spaces7. Canonical Forms of an Operator8. Bilinear and Quadratic Forms9. Sesquilinear and Hermitian Forms10. Applications of Linear Algebra to Numerical Methods11. Affine and Euclidean Spaces and the Applications of Linear Algebra to Geometry12. Ordinary differential equations and linear systems of ordinary differential equations

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Book SynopsisThis third of the three-volume book is targeted as a basic course in algebraic topology and topology for fiber bundles for undergraduate and graduate students of mathematics. It focuses on many variants of topology and its applications in modern analysis, geometry, and algebra. Topics covered in this volume include homotopy theory, homology and cohomology theories, homotopy theory of fiber bundles, Euler characteristic, and the Betti number. It also includes certain classic problems such as the Jordan curve theorem along with the discussions on higher homotopy groups and establishes links between homotopy and homology theories, axiomatic approach to homology and cohomology as inaugurated by Eilenberg and Steenrod. It includes more material than is comfortably covered by beginner students in a one-semester course. Students of advanced courses will also find the book useful. This book will promote the scope, power and active learning of the subject, all the while covering a wide range of theory and applications in a balanced unified way.Table of Contents1. Prerequisite Concepts of Topology, Algebra and Category Theory.- 2. Homotopy Theory: Fundamental and Higher Homotopy Groups.- 3. Homology and Cohomology Theories: An Axiomatic Approach with Consequences.- 4. Topology of Fiber Bundles.- 5. Homotopy Theory of Bundles.- 6. Some Applications of Algebraic Topology.- 7. Brief History on Algebraic Topology and Fiber Bundles.

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Book SynopsisAlgebra is like a foreign language for many students. They have difficulty translating the words, their definitions and how it applies to problem-solving. Tussy/Gustafson''s ELEMENTARY AND INTERMEDIATE ALGEBRA, 6th Edition, addresses these concerns, giving you the tools needed to understand the language of algebra. Strategy and Why explanations in the worked examples show the how and the why behind problem-solving. Algebra is not just about the x -- it''s also about the WHY. The text contains many opportunities to apply the algebraic skills you have learned to solve a wide variety of interesting real-life applications using a six-step problem-solving strategy. In combination, the text and WebAssign will guide you through an integrated learning process that will expand your reasoning abilities as it teaches you how to read, write and think mathematically using the language of Algebra.

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Book SynopsisIn this book, I. Ya. Bakel'man introduces inversion transformations in the Euclidean plane and discusses the interrelationships among more general mathematical concepts. The author begins by defining and giving examples of the concept of a transformation in the Euclidean plane, and then explains the point of infinity and the stereographic projection of the sphere onto the plane. With this preparation, the student is capable of applying the theory of inversions to classical construction problems in the plane. The author also discusses the theory of pencils of circles, and he uses the acquired techniques in a proof of Ptolemy's theorem. In the final chapter, the idea of a group is introduced with applications of group theory to geometry. The author demonstrates the group-theoretic basis for the distinction between Euclidean and Lobachevskian geometry.

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Book SynopsisAn introduction to the calculus, with an excellent balance between theory and technique. Integration is treated before differentiation -- this is a departure from most modern texts, but it is historically correct, and it is the best way to establish the true connection between the integral and the derivative. Proofs of all the important theorems are given, generally preceded by geometric or intuitive discussion. This Second Edition introduces the mean-value theorems and their applications earlier in the text, incorporates a treatment of linear algebra, and contains many new and easier exercises. As in the first edition, an interesting historical introduction precedes each important new concept.Table of ContentsLinear Analysis. Linear Spaces. Linear Transformations and Matrices. Determinants. Eigenvalues and Eigenvectors. Eigenvalues of Operators Acting on Euclidean Spaces. Linear Differential Equations. Systems of Differential Equations. Nonlinear Analysis. Differential Calculus of Scalar and Vector Fields. Applications of the Differential Calculus. Line Integrals. Special Topics. Set Functions and Elementary Probability. Calculus of Probabilities. Introduction to Numerical Analysis.

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Book SynopsisEssential mathematical tools for the study of modern quantum theory. Linear Algebra for Quantum Theory offers an excellent survey of those aspects of set theory and the theory of linear spaces and their mappings that are indispensable to the study of quantum theory.Table of ContentsElements of Set Theory. Linear Spaces. Binary Product Spaces. Axioms of Quantum Theory Formulated as a Trace Algebra. References. Appendices. Index.

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Book SynopsisThrough its presentation of the essentials, this text is briefer than some and has been carefully edited and designed to meet the specific needs of a one-semester course at the appropriate level for a Senior.Following extensive student testing for readability and understandability, examples have been intermixed with the theory throughout the book to introduce, motivate, and extend the main text. Although readability is emphasized, proofs are provided to promote logical thinking. Finally, the author''s conversational style holds the reader''s interest while exploring several important topics that traditionally have been reserved for graduate courses. The result is that students can apply theory that is sometimes a sterile subject in other courses, and can hit the ground running in advanced courses in feedback control design, dynamics of power systems, communications, and signal processing.Trade Review"...an excellent collection of examples...a good addition to the field of linear systems..." (Int Jnl of Robust & Nonlinear Control, May 2002)Table of ContentsSolution of State-Space Equations. Transform Methods. Writing State-Space Equations. Matrices Over a Field. Vector Spaces. Similarity Transformations. Stability. Minimality via Similarity Transformations. Poles and Zeros. References. Appendix. Index.

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Book SynopsisAn accessible and practical introduction to wavelets With applications in image processing, audio restoration, seismology, and elsewhere, wavelets have been the subject of growing excitement and interest over the past several years. Unfortunately, most books on wavelets are accessible primarily to research mathematicians.Trade Review...this text has the advantage of being self contained: the conceptual prerequisites are kept to a minimum and the student can quickly get to the applications with a reasonably complete understanding of how and why wavelets work in data processing. As such, this text should appeal to instructors and students who are interested in learning about how wavelets work without being required to learn the whole mathematical backdrop in which wavelet theory is developed. (Zentralblatt Math, Volume 940, No 15, 2000)Table of ContentsWavelets, Fingerprints, and Image Processing. Wavelets and Orthogonal Decompositions. Multiresolutions, Cascades, and Filters. Sample Projects. Appendices. References. Index.

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Book SynopsisMultivariable Mathematics combines linear algebra and multivariable calculus in a rigorous approach. The material is integrated to emphasize the role of linearity in all of calculus and the recurring theme of implicit versus explicit that persists in linear algebra and analysis. In the text, the author addresses all of the standard computational material found in the usual linear algebra and multivariable calculus courses, and more, interweaving the material as effectively as possible and also including complete proofs. By emphasizing the theoretical aspects and reviewing the linear algebra material quickly, the book can also be used as a text for an advanced calculus or multivariable analysis course culminating in a treatment of manifolds, differential forms, and the generalized Stokes's Theorem.Table of ContentsPreface. Chapter 1. Vectors and Matrices. 1.1 Vectors in Rn.. 1.2 Dot Product. 1.3 Subspaces of Rn. 1.4 Linear Transformations and Matrix Algebra. 1.5 Introduction to Determinates and the Cross Product. Chapter 2. Functions, Limits, and Continuity. 2.1. Scalar- and Vector-Valued Functions. 2.2. A Bit of Topology in Rn. 2.3. Limits and Continuity. Chapter 3. The Derivative. 3.1. Partial Derivatives and Directional Derivatives. 3.2. Differentiability. 3.3. Differentiation Rules. 3.4. The Gradient. 3.5. Curves. 3.6. Higher-Order Partial Derivatives. Chapter 4. Implicit and Explicit Solutions of Linear Systems. 4.1. Gaussian Elimination and the Theory of Linear Systems. 4.2. Elementary Matrices and Calculating Inverse Matrices. 4.3. Linear Independence, Basis, and Dimension. 4.4. The Four Fundamental Subspaces. 4.5. The Nonlinear Case: Introduction to Manifolds. Chapter 5. Extremum Problems. 5.1. Compactness and the Maximum Value Theorem. 5.2. Maximum/Minimum Problems. 5.3. Quadratic Forms and the Second Derivative Test. 5.4. Lagrange Multipliers. 5.5. Projections, Least Squares, and Inner Product Spaces. Chapter 6. Solving Nonlinear Problems. 6.1. The Contraction Mapping Principle. 6.2. The Inverse and Implicit Function Theorems. 6.3. Manifolds Revisited. Chapter 7. Integration. 7.1. Multiple Integrals. 7.2. Iterated Integrals and Fubini’s Theorem. 7.3. Polar, Cylindrical, and Spherical Coordinates. 7.4. Physical Applications. 7.5. Determinants and n-Dimensional Volume. 7.6. Change of Variables Theorem. Chapter 8. Differential Forms and Integration on Manifolds. 8.1. Motivation. 8.2. Differential Forms. 8.3. Line Integrals and Green’s Theorem. 8.4. Surface Integrals and Flux. 8.5. Stokes’s Theorem. 8.6. Applications to Physics. 8.7. Applications to Topology. 9. Eigenvalues, Eigenvectors, and Applications. 9.1. Linear Transformations and Change of Basis. 9.2. Eigenvalues, Eigenvectors, and Diagonalizability. 9.3. Difference Equations and Ordinary Differential Equations. 9.4. The Spectral Theorem. Glossary of Notations and Results from Single-Variable Calculus. For Further Reading. Answers to Selected Exercises. Index.

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Book SynopsisManystudents find it difficult to master the fundamental skills that are essential to succeeding in physics. Now with this helpful book, they''ll quickly learn how to break physics down into basic steps. Author Stuart Loucks presents the material in a way that will motivate and empower them. He offers clear explanations of key concepts while examining the fundamental topics and approaches needed to solve algebra-based physics problems. Understand the basic language of physics Introductory Physics with Algebra as a Second Language will help you make sense of your textbook and class notes so that you can use them more effectively. The text explains key topics in algebra-based physics in clear, easy-to-understand language. Break problems down into simple steps Introductory Physics with Algebra as a Second Language teaches you to recognize details that tell you how to begin new problems. You will learn how to effectively organize thTable of ContentsCHAPTER 1 THE BOTTOM LINE FOR SOLVING PHYSICS PROBLEMS 1 CHAPTER 2 LINEAR VELOCITY AND ACCELERATION 3 2.1 Linear Motion Equations 3 2.2 The Idea Behind How to Use Motion Equations 4 2.3 Constant/Average Speed or Velocity Problems 5 2.4 Constant/Average Speed or Velocity—Two Intervals, Same Direction 6 2.5 Constant/Average Speed or Velocity—Two Intervals, Direction Change 10 2.6 Constant/Average Speed or Velocity—Two Objects 13 2.7 How to Set Up Constant/Average Speed or Velocity Problems 15 2.8 Constant/Average Acceleration Problems 16 2.9 Constant/Average Acceleration—One Interval 17 2.10 Constant/Average Acceleration—Multiple Intervals 20 2.11 Constant/Average Acceleration—“Free-Fall” 22 2.12 Constant/Average Acceleration—Two Objects 27 2.13 How to Set Up Constant/Average Acceleration Problems 29 CHAPTER 3 VECTORS 31 3.1 Magnitude and Direction, and x- and y-Components 31 3.2 Vectors along One Axis 37 3.3 Vector Addition 38 3.4 How to Set Up Vector Problems 42 3.5 “Back Where You Started”—When Vectors Add to Zero 42 3.6 Subtracting Vectors, OR, When One of the Added Vectors Is Unknown 45 CHAPTER 4 PROJECTILE MOTION 49 4.1 Projectile Motion: Combining Three Basic Concepts 49 4.2 When Initial Velocity Is Horizontal 50 4.3 How to Set Up Projectile Motion Problems 55 4.4 When Final Velocity Is Horizontal (at Maximum Height) 56 4.5 When Initial and Final Heights Are Equal 61 4.6 When Both Initial and Final Velocities Are at Angles 62 CHAPTER 5 FORCE AND NEWTON’S LAWS OF MOTION 69 5.1 How to Draw a Free-Body Diagram (FBD) 70 5.2 Forces in 1D 72 5.3 How to Set Up Force Problems 75 5.4 Motion Intervals in Force Problems 76 5.5 Objects Connected by Strings, Ropes, and so on 80 5.6 Forces in 2D 83 5.7 Sliding—Kinetic Friction 86 5.8 “Just about to Slip”—Maximum Static Friction 90 5.9 Inclines or Ramps 93 5.10 Objects Pushing on Each Other 97 CHAPTER 6 CIRCULAR MOTION AND CENTRIPETAL FORCE 100 6.1 Tangential Speed and Centripetal Acceleration 100 6.2 Comparing Circular Motion at Two Different Radii 101 6.3 Comparing Circular Motion at Two Different Speeds 102 6.4 How to Set Up Circular Motion Comparison Problems 103 6.5 How to Think about Centripetal Force Problems 104 6.6 Circular Motion with a Horizontal String 105 6.7 How to Set Up Centripetal Force Problems 107 6.8 Circular Motion with a String at an Angle 107 6.9 Circular Motion on an Unbanked Road with Friction 111 6.10 Circular Motion on a Banked Road without Friction 113 6.11 Vertical Circular Motion—Lowest Point 117 6.12 Vertical Circular Motion—Highest Point, Upside-Down 118 6.13 Vertical Circular Motion—Highest Point, Right-Side-Up 121 CHAPTER 7 GRAVITATION AND ORBITS 123 7.1 Weight and g at a Planet’s Surface 123 7.2 Adding Gravitational Force Vectors 125 7.3 Circular Orbit Problems 129 7.4 Circular Orbit Equations 133 7.5 Comparing Orbits at Two Different Radii 134 CHAPTER 8 WORK AND ENERGY 136 8.1 Work Done by a Constant/Average Force 136 8.2 Work Problems—with Two or More Forces 137 8.3 Work Problems—when Forces Are Not Given 140 8.4 How to Set Up Work Problems 147 8.5 The Work-Energy Theorem—KE Only 148 8.6 How to Set Up Work-Energy Problems—KE only 153 8.7 Potential Energy, Conservative and Nonconservative Forces 154 8.8 The Work-Energy Theorem—KE and PE 155 8.9 How to Set Up Work-Energy Problems—KE and PE 161 8.10 Conservation of Energy—When Wnc = 0 162 8.11 How to Set Up Conservation of Energy Problems 166 8.12 How to Split Up a Difficult Problem 166 CHAPTER 9 IMPULSE, MOMENTUM, AND CENTER OF MASS 170 9.1 The Impulse-Momentum Theorem 170 9.2 1D Impulse and Momentum 172 9.3 2D Impulse and Momentum 174 9.4 How to Set Up Impulse and Momentum Problems 178 9.5 Conservation of Momentum 179 9.6 1D Collisions—Objects Coming Together 181 9.7 1D Explosions—Objects Pushing Apart 183 9.8 1D Elastic Collisions 185 9.9 2D Collisions 189 9.10 How to Set Up Conservation of Momentum Problems 195 9.11 Center of Mass 196 9.12 1D Center of Mass 196 9.13 2D Center of Mass 198 9.14 How to Set Up Center of Mass Problems 201 CHAPTER 10 ANGULAR VELOCITY AND ACCELERATION 202 10.1 How to Relate Angular and Tangential or Linear Quantities 202 10.2 Two-Object, Two-Circle Problems 204 10.3 How to Set up two-Object, Two-Circle Problems 208 10.4 Constant/Average Angular Velocity 209 10.5 How to Set Up Constant/Average Angular Velocity Problems 212 10.6 Constant/Average Angular Acceleration 212 10.7 Constant/Average Angular Acceleration—Multiple Intervals 215 10.8 Constant/Average Angular Acceleration—with Tangential or Linear Acceleration 218 10.9 Constant/Average Angular Acceleration—with Centripetal Acceleration 223 10.10 Summary of Angular Velocity and Acceleration Equations 226 10.11 How to Set Up Constant/Average Angular Acceleration Problems 227 CHAPTER 11 TORQUE AND EQUILIBRIUM 228 11.1 Torque 228 11.2 How to Set Up Torque Problems 231 11.3 Equilibrium for “Rigid” Bodies 232 11.4 Equilibrium—With Only 90° Angles 232 11.5 Equilibrium—With Non-90° Angles 245 11.6 How to Set Up Equilibrium Problems 251 CHAPTER 12 MORE ANGULAR MOTION 252 12.1 Moment of Inertia 252 12.2 Torque and Angular Acceleration Problems 254 12.3 How to Set Up Torque and Angular Acceleration Problems 259 12.4 Rotational Kinetic Energy and Conservation of Energy 259 12.5 Conservation of Angular Momentum 264 12.6 Conservation of Angular Momentum Problems—First Type 265 12.7 Conservation of Angular Momentum Problems—Second Type 268 12.8 How to Set Up Conservation of Angular Momentum Problems 271 INDEX 273

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Book SynopsisDetails the basic theory of polynomial and fractional representation methods for algebraic analysis and synthesis of linear multivariable control systems. It also serves as a self-contained treatise of the mathematical theory so that results and techniques of the ``state space approaches'''' for regular and singular systems appear as special cases of a general theory covering the wider class of PMDs of linear systems. Among the topics covered are: real rational vector spaces and rational matrices, pole and zero structure of rational matrices at infinity, proper and omega stable rational fuctions and matrices.Table of ContentsReal Rational Vector Spaces and Rational Matrices. Polynomial Matrix Models of Linear Multivariable Systems. Pole and Zero Structure of Rational Matrices at Infinity. Dynamics of Polynomial Matrix Models. Proper and Omega-Stable Rational Functions and Matrices. Feedback System Stability and Stabilization. Some Algebraic Design Problems. Notations. Appendices. Index.

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Book SynopsisQuarternionic calculus covers a branch of mathematics which uses computational techniques to help solve problems from a wide variety of physical systems which are mathematically modelled in 3, 4 or more dimensions. Examples of the application areas include thermodynamics, hydrodynamics, geophysics and structural mechanics. Focusing on the Clifford algebra approach the authors have drawn together the research into quarternionic calculus to provide the non-expert or research student with an accessible introduction to the subject. This book fills the gap between the theoretical representations and the requirements of the user.Table of ContentsQuanternions and Multivectors. Clifford Valued Functions and Forms. Clifford Operator Calculus. Boundary Value Problems. Numerical Clifford Analysis. Further Results and Research Problems. Appendices. Index.

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Book SynopsisMatrix Differential Calculus With Applications in Statistics and Econometrics Revised Edition Jan R. Magnus, CentER, Tilburg University, The Netherlands and Heinz Neudecker, Cesaro, Schagen, The Netherlands .deals rigorously with many of the problems that have bedevilled the subject up to the present time. - Stephen Pollock, Econometric Theory I continued to be pleasantly surprised by the variety and usefulness of its contents - Isabella Verdinelli, Journal of the American Statistical Association Continuing the success of their first edition, Magnus and Neudecker present an exhaustive and self-contained revised text on matrix theory and matrix differential calculus. Matrix calculus has become an essential tool for quantitative methods in a large number of applications, ranging from social and behavioural sciences to econometrics. While the structure and successful elements of the first edition remain, this revised and updated edition contains many new examples and exercises. * CoTrade Review"...the best book to learn matrix and related ideas...statisticians, econometricians, computer scientists, engineers, and psychometricians will find this extremely useful." (Journal of Statistical Computation and Simulation, March 2006) "a most welcome revision" (Computational Statistics & Data Analysis, 28 August 2001)Table of ContentsPreface xv Preface to the first revised printing xvii Preface to the second revised printing xviii Part One- Matrices Part Two- Differentials: the theory Part Three- Differentials: the practice Part Four- Inequalities Part Five- The linear model Part Six- Applications to maximum likelihood estimation Bibliography 379 Index of Symbols 387 Subject Index 390

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Book SynopsisElliptic Tales describes the latest developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematics--the Birch and Swinnerton-Dyer Conjecture. In this book, Avner Ash and Robert Gross guide readers through the mathematics they need to understand this captivating problem. The key to the conjectureTrade Review"The authors present their discussion in an informal, sometimes playful manner and with detail that will appeal to an audience with a basic understanding of calculus. This book will captivate math enthusiasts as well as readers curious about an intriguing and still unanswered question."--Margaret Dominy, Library Journal "Minimal prerequisites and its clear writing make this book (which even has a few exercises) a great choice for a seminar for mathematics majors, who at some point should have such an excursion to one of the frontiers of mathematics."--Mathematics Magazine "The authors of Elliptic Tales do a superb job in demonstrating the approach that mathematicians take when they confront unsolved problems involving elliptic curves."--Sungkon Chang, Times Higher Education "One cannot help being impressed, in reading the book and pursuing a few of the references, by the magnitude of the enterprise it chronicles."--James Case, SIAM News "Ash and Gross thoroughly explain the statement and significance of the linchpin Birch and Swinnerton-Dyer conjection... [A]sh and Gross deliver ample and current intellectual and technical substance."--Choice "I would envision this book as an excellent text for an undergraduate 'capstone' course in mathematics; the book lends itself to independent reading, but topics may be explored in much greater depth and rigor in the classroom. Additionally, the book indeed brings together ideas from calculus, complex variables and algebra, showing how a single mathematical research question may require an integrated understanding of the various branches of mathematics. Thus, it encourages students to reinforce their understanding of these various fields, while simultaneously introducing them to an open question in mathematics and a vibrant field of study."--Lisa A. Berger, Mathematical Reviews Clippings "The book is very pleasantly written, and in my opinion, the authors have done an admirable job in giving an idea to non-experts what the Birch-Swinnerton Dyer conjecture is about."--Jan-Hendrik Evertse, Zentralblatt MATH "The book's most important contributions ... are the sense of discovery, invention, and insight into the habits of mind used by mathematicians on this journey. I would recommend this book to anyone who wants to be challenged mathematically or who wants to experience mathematics as creative and exciting."--Jacqueline Coomes, Mathematics Teacher "[T]his book is a wonderful introduction to what is arguably one of the most important mathematical problems of our time and for that reason alone it deserves to be widely read. Another reason to recommend this book is the opportunity to share in the readily apparent joy the authors have for their subject and the beauty they see in it, not least because ... joy and beauty are the most important reasons for doing mathematics, irrespective of its dollar value."--Rob Ashmore, Mathematics Today "This book has many nice aspects. Ash and Gross give a truly stimulating introduction to elliptic curves and the BSD conjecture for undergraduate students. The main achievement is to make a relative easy exposition of these so technical topics."--Jonathan Sanchez-Hernandez, Mathematical SocietyTable of ContentsPreface xiii Acknowledgments xix Prologue 1 PART I. DEGREE Chapter 1. Degree of a Curve 13 1.Greek Mathematics 13 2.Degree 14 3.Parametric Equations 20 4.Our Two Definitions of Degree Clash 23 Chapter 2. Algebraic Closures 26 1.Square Roots of Minus One 26 2.Complex Arithmetic 28 3.Rings and Fields 30 4.Complex Numbers and Solving Equations 32 5.Congruences 34 6.Arithmetic Modulo a Prime 38 7.Algebraic Closure 38 Chapter 3. The Projective Plane 42 1.Points at Infinity 42 2.Projective Coordinates on a Line 46 3.Projective Coordinates on a Plane 50 4.Algebraic Curves and Points at Infinity 54 5.Homogenization of Projective Curves 56 6.Coordinate Patches 61 Chapter 4. Multiplicities and Degree 67 1.Curves as Varieties 67 2.Multiplicities 69 3.Intersection Multiplicities 72 4.Calculus for Dummies 76 Chapter 5. B'ezout's Theorem 82 1.A Sketch of the Proof 82 2.An Illuminating Example 88 PART II. ELLIPTIC CURVES AND ALGEBRA Chapter 6. Transition to Elliptic Curves 95 Chapter 7. Abelian Groups 100 1.How Big Is Infinity? 100 2.What Is an Abelian Group? 101 3.Generations 103 4.Torsion 106 5.Pulling Rank 108 Appendix: An Interesting Example of Rank and Torsion 110 Chapter 8. Nonsingular Cubic Equations 116 1.The Group Law 116 2.Transformations 119 3.The Discriminant 121 4.Algebraic Details of the Group Law 122 5.Numerical Examples 125 6.Topology 127 7.Other Important Facts about Elliptic Curves 131 5.Two Numerical Examples 133 Chapter 9. Singular Cubics 135 1.The Singular Point and the Group Law 135 2.The Coordinates of the Singular Point 136 3.Additive Reduction 137 4.Split Multiplicative Reduction 139 5.Nonsplit Multiplicative Reduction 141 6.Counting Points 145 7.Conclusion 146 Appendix A: Changing the Coordinates of the Singular Point 146 Appendix B: Additive Reduction in Detail 147 Appendix C: Split Multiplicative Reduction in Detail 149 Appendix D: Nonsplit Multiplicative Reduction in Detail 150 Chapter 10. Elliptic Curves over Q 152 1.The Basic Structure of the Group 152 2.Torsion Points 153 3.Points of Infinite Order 155 4.Examples 156 PART III. ELLIPTIC CURVES AND ANALYSIS Chapter 11. Building Functions 161 1.Generating Functions 161 2.Dirichlet Series 167 3.The Riemann Zeta-Function 169 4.Functional Equations 171 5.Euler Products 174 6.Build Your Own Zeta-Function 176 Chapter 12. Analytic Continuation 181 1.A Difference that Makes a Difference 181 2.Taylor Made 185 3.Analytic Functions 187 4.Analytic Continuation 192 5.Zeroes, Poles, and the Leading Coefficient 196 Chapter 13. L-functions 199 1.A Fertile Idea 199 2.The Hasse-Weil Zeta-Function 200 3.The L-Function of a Curve 205 4.The L-Function of an Elliptic Curve 207 5.Other L-Functions 212 Chapter 14. Surprising Properties of L-functions 215 1.Compare and Contrast 215 2.Analytic Continuation 220 3.Functional Equation 221 Chapter 15. The Conjecture of Birch and Swinnerton-Dyer 225 1.How Big Is Big? 225 2.Influences of the Rank on the Np's 228 3.How Small Is Zero? 232 4.The BSD Conjecture 236 5.Computational Evidence for BSD 238 6.The Congruent Number Problem 240 Epilogue 245 Retrospect 245 Where DoWe Go from Here? 247 Bibliography 249 Index 251

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Book SynopsisTrade Review"A Choice Outstanding Academic Title of the Year"

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Book SynopsisTrade Review"This book is without a doubt the most enjoyable, stimulating book of mathematical physics (and occasionally more pure branches of maths) puzzles that I have ever read. It’s essentially a series of cleverly, and occasionally fiendishly put-together mathematics and physics challenge questions, each of which gets you thinking in a new and fascinating way."---Jonathan Shock, Mathemafrica"Reading Nahin is like reading through a select library of ancient Babylonian mathematical clay tablets. Surprises abound. . . . Nahin weaves much colorful history into his narrative."---Andrew Simoson, Mathematical Intelligencer"Engaging. . . . The book contains a wealth of original problems. . . . An enjoyable read."---Antonín Slavík, Zentralblatt MATH"This reviewer found himself being drawn to a variety of unfamiliar settings with much interest and even fascination." * Choice *"I certainly enjoyed [the book]!"---Alan Stevens, Mathematics Today"The potential audience for this book should be fairly large and go from highly talented high school students up through professionals in any STEM field."---Geoffrey Dietz, MAA Reviews

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Book SynopsisPraise for the Third Edition . . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . .Zentralblatt MATH The Fourth Edition of Introduction to Abstract Algebra continues to provide an accessible approach to the basic structures of abstract algebra: groups, rings, and fields. The book''s unique presentation helps readers advance to abstract theory by presenting concrete examples of induction, number theory, integers modulo n, and permutations before the abstract structures are defined. Readers can immediately begin to perform computations using abstract concepts that are developed in greater detail later in the text. The Fourth Edition features important concepts as well as specialized topics, including: The treatment of nilpotent groups, including the Frattini and Fitting subgroups Symmetric polynomials The pTrade Review “This could also be an excellent adjunct to more theoretically oriented textbooks used in more intensive courses.” (Computing Reviews, 5 November 2012) Table of ContentsPREFACE ix ACKNOWLEDGMENTS xvii NOTATION USED IN THE TEXT xix A SKETCH OF THE HISTORY OF ALGEBRA TO 1929 xxiii 0 Preliminaries 1 0.1 Proofs / 1 0.2 Sets / 5 0.3 Mappings / 9 0.4 Equivalences / 17 1 Integers and Permutations 23 1.1 Induction / 24 1.2 Divisors and Prime Factorization / 32 1.3 Integers Modulo n / 42 1.4 Permutations / 53 1.5 An Application to Cryptography / 67 2 Groups 69 2.1 Binary Operations / 70 2.2 Groups / 76 2.3 Subgroups / 86 2.4 Cyclic Groups and the Order of an Element / 90 2.5 Homomorphisms and Isomorphisms / 99 2.6 Cosets and Lagrange’s Theorem / 108 2.7 Groups of Motions and Symmetries / 117 2.8 Normal Subgroups / 122 2.9 Factor Groups / 131 2.10 The Isomorphism Theorem / 137 2.11 An Application to Binary Linear Codes / 143 3 Rings 159 3.1 Examples and Basic Properties / 160 3.2 Integral Domains and Fields / 171 3.3 Ideals and Factor Rings / 180 3.4 Homomorphisms / 189 3.5 Ordered Integral Domains / 199 4 Polynomials 202 4.1 Polynomials / 203 4.2 Factorization of Polynomials Over a Field / 214 4.3 Factor Rings of Polynomials Over a Field / 227 4.4 Partial Fractions / 236 4.5 Symmetric Polynomials / 239 4.6 Formal Construction of Polynomials / 248 5 Factorization in Integral Domains 251 5.1 Irreducibles and Unique Factorization / 252 5.2 Principal Ideal Domains / 264 6 Fields 274 6.1 Vector Spaces / 275 6.2 Algebraic Extensions / 283 6.3 Splitting Fields / 291 6.4 Finite Fields / 298 6.5 Geometric Constructions / 304 6.6 The Fundamental Theorem of Algebra / 308 6.7 An Application to Cyclic and BCH Codes / 310 7 Modules over Principal Ideal Domains 324 7.1 Modules / 324 7.2 Modules Over a PID / 335 8 p-Groups and the Sylow Theorems 349 8.1 Products and Factors / 350 8.2 Cauchy’s Theorem / 357 8.3 Group Actions / 364 8.4 The Sylow Theorems / 371 8.5 Semidirect Products / 379 8.6 An Application to Combinatorics / 382 9 Series of Subgroups 388 9.1 The Jordan–H¨older Theorem / 389 9.2 Solvable Groups / 395 9.3 Nilpotent Groups / 401 10 Galois Theory 412 10.1 Galois Groups and Separability / 413 10.2 The Main Theorem of Galois Theory / 422 10.3 Insolvability of Polynomials / 434 10.4 Cyclotomic Polynomials and Wedderburn’s Theorem / 442 11 Finiteness Conditions for Rings and Modules 447 11.1 Wedderburn’s Theorem / 448 11.2 The Wedderburn–Artin Theorem / 457 Appendices 471 Appendix A Complex Numbers / 471 Appendix B Matrix Algebra / 478 Appendix C Zorn’s Lemma / 486 Appendix D Proof of the Recursion Theorem / 490 BIBLIOGRAPHY 492 SELECTED ANSWERS 495 INDEX 523

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Book SynopsisThis Student Solutions Manual to Accompany Linear Algebra: Ideas and Applications, Fourth Edition contains solutions to the odd numbered problems to further aid in reader comprehension, and an Instructor''s Solutions Manual (inclusive of suggested syllabi) is available via written request to the Publisher. Both the Student and Instructor Manuals have been enhanced with further discussions of the applications sections, which is ideal for readers who wish to obtain a deeper knowledge than that provided by pure algorithmic approaches. Linear Algebra: Ideas and Applications, Fourth Edition provides a unified introduction to linear algebra while reinforcing and emphasizing a conceptual and hands-on understanding of the essential ideas. Promoting the development of intuition rather than the simple application of methods, this book successfully helps readers to understand not only how to implement a technique, but why its use is important.Table of ContentsSTUDENT MANUAL 1 1 SYSTEMS OF LINEAR EQUATIONS 3 1.1 The Vector Space of m × n Matrices / 3 1.1.2 Applications to Graph Theory I / 7 1.2 Systems / 8 1.2.2 Applications to Circuit Theory / 11 1.3 Gaussian Elimination / 13 1.3.2 Applications to Traffic Flow / 18 1.4 Column Space and Nullspace / 19 2 LINEAR INDEPENDENCE AND DIMENSION 26 2.1 The Test for Linear Independence / 26 2.2 Dimension / 33 2.2.2 Applications to Differential Equations / 37 2.3 Row Space and the Rank-Nullity Theorem / 38 3 LINEAR TRANSFORMATIONS 43 3.1 The Linearity Properties / 43 3.2 Matrix Multiplication (Composition) / 49 3.2.2 Applications to Graph Theory II / 55 3.3 Inverses / 55 3.3.2 Applications to Economics / 60 3.4 The LU Factorization / 61 3.5 The Matrix of a Linear Transformation / 62 4 DETERMINANTS 67 4.1 Definition of the Determinant / 67 4.2 Reduction and Determinants / 69 4.2.1 Volume / 72 4.3 A Formula for Inverses / 74 5 EIGENVECTORS AND EIGENVALUES 76 5.1 Eigenvectors / 76 5.1.2 Application to Markov Processes / 79 5.2 Diagonalization / 80 5.2.1 Application to Systems of Differential Equations / 82 5.3 Complex Eigenvectors / 83 6 ORTHOGONALITY 85 6.1 The Scalar Product in ℝn / 85 6.2 Projections: The Gram-Schmidt Process / 87 6.3 Fourier Series: Scalar Product Spaces / 89 6.4 Orthogonal Matrices / 92 6.5 Least Squares / 93 6.6 Quadratic Forms: Orthogonal Diagonalization / 94 6.7 The Singular Value Decomposition (SVD) / 97 6.8 Hermitian Symmetric and Unitary Matrices / 98 7 GENERALIZED EIGENVECTORS 100 7.1 Generalized Eigenvectors / 100 7.2 Chain Bases / 104 8 NUMERICAL TECHNIQUES 107 8.1 Condition Number / 107 8.2 Computing Eigenvalues / 108

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Book SynopsisA thoroughly updated guide to matrix algebra and it uses in statistical analysis and features SAS, MATLAB, and R throughout This Second Edition addresses matrix algebra that is useful in the statistical analysis of data as well as within statistics as a whole. The material is presented in an explanatory style rather than a formal theorem-proof format and is self-contained. Featuring numerous applied illustrations, numerical examples, and exercises, the book has been updated to include the use of SAS, MATLAB, and R for the execution of matrix computations. In addition, André I. Khuri, who has extensive research and teaching experience in the field, joins this new edition as co-author. The Second Edition also: Contains new coverage on vector spaces and linear transformations and discusses computational aspects of matrices Covers the analysis of balanced linear models using direct products of matrices Analyzes multiresponse linTrade Review"Matrix Algebra Useful for Statistics, Second Edition is an ideal textbook for advanced undergraduate and first-year graduate level courses in statistics and other related disciplines. The book is also appropriate as a reference for independent readers who use statistics and wish to improve their knowledge of matrix algebra." Mathematical Reviews, Sept 2017 Table of ContentsPREFACE xvii PREFACE TO THE FIRST EDITION xix INTRODUCTION xxi ABOUT THE COMPANION WEBSITE xxxi PART I DEFINITIONS, BASIC CONCEPTS, AND MATRIX OPERATIONS 1 1 Vector Spaces, Subspaces, and Linear Transformations 3 1.1 Vector Spaces 3 1.2 Base of a Vector Space 5 1.3 Linear Transformations 7 2 Matrix Notation and Terminology 11 2.1 Plotting of a Matrix 14 2.2 Vectors and Scalars 16 2.3 General Notation 16 3 Determinants 21 3.1 Expansion by Minors 21 3.2 Formal Definition 25 3.3 Basic Properties 27 3.4 Elementary Row Operations 34 3.5 Examples 37 3.6 Diagonal Expansion 39 3.7 The Laplace Expansion 42 3.8 Sums and Differences of Determinants 44 3.9 A Graphical Representation of a 3 × 3 Determinant 45 4 Matrix Operations 51 4.1 The Transpose of a Matrix 51 4.2 Partitioned Matrices 52 4.3 The Trace of a Matrix 55 4.4 Addition 56 4.5 Scalar Multiplication 58 4.6 Equality and the Null Matrix 58 4.7 Multiplication 59 4.8 The Laws of Algebra 74 4.9 Contrasts With Scalar Algebra 76 4.10 Direct Sum of Matrices 77 4.11 Direct Product of Matrices 78 4.12 The Inverse of a Matrix 80 4.13 Rank of a Matrix—Some Preliminary Results 82 4.14 The Number of LIN Rows and Columns in a Matrix 84 4.15 Determination of the Rank of a Matrix 85 4.16 Rank and Inverse Matrices 87 4.17 Permutation Matrices 87 5 Special Matrices 97 5.1 Symmetric Matrices 97 5.2 Matrices Having All Elements Equal 102 5.3 Idempotent Matrices 104 5.4 Orthogonal Matrices 106 5.5 Parameterization of Orthogonal Matrices 109 5.6 Quadratic Forms 110 5.7 Positive Definite Matrices 113 6 Eigenvalues and Eigenvectors 119 6.1 Derivation of Eigenvalues 119 6.2 Elementary Properties of Eigenvalues 122 6.3 Calculating Eigenvectors 125 6.4 The Similar Canonical Form 128 6.5 Symmetric Matrices 131 6.6 Eigenvalues of Orthogonal and Idempotent Matrices 135 6.7 Eigenvalues of Direct Products and Direct Sums of Matrices 138 6.8 Nonzero Eigenvalues of AB and BA 140 7 Diagonalization of Matrices 145 7.1 Proving the Diagonability Theorem 145 7.2 Other Results for Symmetric Matrices 148 7.3 The Cayley–Hamilton Theorem 152 7.4 The Singular-Value Decomposition 153 8 Generalized Inverses 159 8.1 The Moore–Penrose Inverse 159 8.2 Generalized Inverses 160 8.3 Other Names and Symbols 164 8.4 Symmetric Matrices 165 9 Matrix Calculus 171 9.1 Matrix Functions 171 9.2 Iterative Solution of Nonlinear Equations 174 9.3 Vectors of Differential Operators 175 9.4 Vec and Vech Operators 179 9.5 Other Calculus Results 181 9.6 Matrices with Elements That Are Complex Numbers 188 9.7 Matrix Inequalities 189 PART II APPLICATIONS OF MATRICES IN STATISTICS 199 10 Multivariate Distributions and Quadratic Forms 201 10.1 Variance-Covariance Matrices 202 10.2 Correlation Matrices 203 10.3 Matrices of Sums of Squares and Cross-Products 204 10.4 The Multivariate Normal Distribution 207 10.5 Quadratic Forms and ;;2-Distributions 208 10.6 Computing the Cumulative Distribution Function of a Quadratic Form 213 11 Matrix Algebra of Full-Rank Linear Models 219 11.1 Estimation of ;; by the Method of Least Squares 220 11.2 Statistical Properties of the Least-Squares Estimator 226 11.3 Multiple Correlation Coefficient 229 11.4 Statistical Properties under the Normality Assumption 231 11.5 Analysis of Variance 233 11.6 The Gauss–Markov Theorem 234 11.7 Testing Linear Hypotheses 237 11.8 Fitting Subsets of the x-Variables 246 11.9 The Use of the R(.|.) Notation in Hypothesis Testing 247 12 Less-Than-Full-Rank Linear Models 253 12.1 General Description 253 12.2 The Normal Equations 256 12.3 Solving the Normal Equations 257 12.4 Expected Values and Variances 259 12.5 Predicted y-Values 260 12.6 Estimating the Error Variance 261 12.7 Partitioning the Total Sum of Squares 262 12.8 Analysis of Variance 263 12.9 The R(⋅|⋅) Notation 265 12.10 Estimable Linear Functions 266 12.11 Confidence Intervals 272 12.12 Some Particular Models 272 12.13 The R(⋅|⋅) Notation (Continued) 277 12.14 Reparameterization to a Full-Rank Model 281 13 Analysis of Balanced Linear Models Using Direct Products of Matrices 287 13.1 General Notation for Balanced Linear Models 289 13.2 Properties Associated with Balanced Linear Models 293 13.3 Analysis of Balanced Linear Models 298 14 Multiresponse Models 313 14.1 Multiresponse Estimation of Parameters 314 14.2 Linear Multiresponse Models 316 14.3 Lack of Fit of a Linear Multiresponse Model 318 PART III MATRIX COMPUTATIONS AND RELATED SOFTWARE 327 15 SAS/IML 329 15.1 Getting Started 329 15.2 Defining a Matrix 329 15.3 Creating a Matrix 330 15.4 Matrix Operations 331 15.5 Explanations of SAS Statements Used Earlier in the Text 354 16 Use of MATLAB in Matrix Computations 363 16.1 Arithmetic Operators 363 16.2 Mathematical Functions 364 16.3 Construction of Matrices 365 16.4 Two- and Three-Dimensional Plots 371 17 Use of R in Matrix Computations 383 17.1 Two- and Three-Dimensional Plots 396 Exercises 408 APPENDIX 413 INDEX 475

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Book SynopsisMathematical models have become a crucial way for the Earth scientist to understand and predict how our planet functions and evolves through time and space. The finite element method (FEM) is a remarkably flexible and powerful tool with enormous potential in the Earth Sciences.Table of ContentsPreface xiii Symbols xv About the Companion Website xvii Part I The Finite Element Method with Matlab 1 1 Preliminaries 3 1.1 Mathematical Models 3 1.2 Boundary and Initial Conditions 4 1.3 Analytical Solutions 5 1.4 Numerical Solutions 5 1.5 Numerical Solution Methods 7 1.6 Matlab Script 8 1.7 Exercises 10 Suggested Reading 12 2 Beginning with the Finite Element Method 13 2.1 The Governing PDE 13 2.2 Approximating the Continuous Variable 14 2.3 Minimizing the Residual 15 2.4 Evaluating the Element Matrices 17 2.5 Time Discretization 18 2.6 Assembly 19 2.7 Boundary and Initial Conditions 21 2.8 Solution of the Algebraic Equations 21 2.9 Exercises 22 Suggested Reading 23 3 Programming the Finite Element Method in Matlab 25 3.1 Program Structure and Philosophy 25 3.2 Summary of the Problem 25 3.3 Discretized Equations 26 3.4 The Program 27 3.4.1 Preprocessor Stage 27 3.4.2 Solution Stage 29 3.4.3 Postprocessor Stage 30 3.5 Matlab Script 30 3.6 Exercises 33 Suggested Reading 34 4 Numerical Integration and Local Coordinates 35 4.1 Gauss–Legendre Quadrature 36 4.2 Local Coordinates 37 4.3 Evaluating the Integrals 39 4.4 Variable Material Properties 40 4.5 Programming Considerations 41 4.6 Matlab Script 43 4.7 Exercises 45 Suggested Reading 47 5 The Finite Element Method in Two Dimensions 49 5.1 Discretization 50 5.2 Geometry and Nodal Connectivity 52 5.3 Integration of Element Matrices 54 5.4 Multielement Assembly 57 5.5 Boundary Conditions and Solution 60 5.6 Matlab Script 61 5.7 Exercises 65 Suggested Reading 66 6 The Finite Element Method in Three Dimensions 67 6.1 Discretization 67 6.2 Element Integration 69 6.3 Assembly for Multielement Mesh 72 6.4 Boundary Conditions and Solution 73 6.5 Matlab Program 74 6.6 Exercises 79 Suggested Reading 80 7 Generalization of Finite Element Concepts 81 7.1 The FEM for an Elliptic Problem 84 7.2 The FEM for a Hyperbolic Problem 96 7.3 The FEM for Systems of Equations 102 7.4 Exercises 116 Suggested Reading 116 Part II Applications of the Finite Element Method in Earth Science 119 8 Heat Transfer 121 8.1 Conductive Cooling in an Eroding Crust 122 8.2 Conductive Cooling of an Intrusion 126 Suggested Reading 135 9 Landscape Evolution 137 9.1 Evolution of a 1D River Profile 138 9.2 Evolution of a Fluvially Dissected Landscape 143 Suggested Reading 150 10 Fluid Flow in Porous Media 151 10.1 Fluid Flow Around a Fault 152 10.2 Viscous Fingering 157 Suggested Reading 166 11 Lithospheric Flexure 167 11.1 Governing Equations 167 11.2 FEM Discretization 168 11.3 Matlab Implementation 171 Suggested Reading 181 12 Deformation of Earth’s Crust 183 12.1 Governing Equations 183 12.2 Rate Formulation 185 12.3 FEM Discretization 186 12.4 Viscoelastoplasticity 188 12.5 Matlab Implementation 190 Suggested Reading 205 13 Going Further 207 13.1 Optimization 207 13.2 Using Other FEMs 213 13.3 Use of Existing Finite Element Software 215 Appendix A Derivation of the Diffusion Equation 217 Appendix B Basics of Linear Algebra with Matlab 221 Appendix C Comparison between Different Numerical Methods 227 Appendix D Integration by Parts 237 Appendix E Time Discretization 239 References 241 Index 245

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Book SynopsisBasic Math & Pre-Algebra For Dummies, 2nd Edition (9781119293637) was previously published as Basic Math & Pre-Algebra For Dummies, 2nd Edition (9781118791981). While this version features a new Dummies cover and design, the content is the same as the prior release and should not be considered a new or updated product. Tips for simplifying tricky basic math and pre-algebra operations Whether you''re a student preparing to take algebra or a parent who wants or needs to brush up on basic math, this fun, friendly guide has the tools you need to get in gear. From positive, negative, and whole numbers to fractions, decimals, and percents, you''ll build necessary math skills to tackle more advanced topics, such as imaginary numbers, variables, and algebraic equations. Explanations and practical examples that mirror today''s teaching methods Relevant cultural vernacular and references Standard For Dummiesmaterials thatTable of ContentsIntroduction 1 About This Book 1 Foolish Assumptions 2 Icons Used in This Book 3 Beyond the Book 3 Where to Go from Here 4 Part 1: Getting Started with Basic Math and Pre-algebra 5 Chapter 1: Playing the Numbers Game 7 Inventing Numbers 8 Understanding Number Sequences 8 Evening the odds 9 Counting by threes, fours, fives, and so on 9 Getting square with square numbers 10 Composing yourself with composite numbers 11 Stepping out of the box with prime numbers 12 Multiplying quickly with exponents 13 Looking at the Number Line 14 Adding and subtracting on the number line 14 Getting a handle on nothing, or zero 15 Taking a negative turn: Negative numbers 16 Multiplying the possibilities 17 Dividing things up 17 Discovering the space in between: Fractions 19 Four Important Sets of Numbers 19 Counting on the counting numbers 20 Introducing integers 20 Staying rational 21 Getting real 21 Chapter 2: It’s All in the Fingers: Numbers and Digits 23 Knowing Your Place Value 23 Counting to ten and beyond 24 Telling placeholders from leading zeros 25 Reading long numbers 26 Close Enough for Rock ’n’ Roll: Rounding and Estimating 26 Rounding numbers 26 Estimating value to make problems easier 28 Table of Contents Chapter 3: The Big Four: Addition, Subtraction, Multiplication, and Division 29 Adding Things Up 30 In line: Adding larger numbers in columns 30 Carry on: Dealing with two-digit answers 31 Take It Away: Subtracting 31 Columns and stacks: Subtracting larger numbers 32 Can you spare a ten? Borrowing to subtract 33 Multiplying 35 Signs of the times 36 Memorizing the multiplication table 37 Double digits: Multiplying larger numbers 40 Doing Division Lickety-Split 42 Making short work of long division 42 Getting leftovers: Division with a remainder 45 Part 2: Getting a Handle on Whole Numbers 47 Chapter 4: Putting the Big Four Operations to Work 49 Knowing Properties of the Big Four Operations 50 Inverse operations 50 Commutative operations 51 Associative operations 52 Distribution to lighten the load 53 Doing Big Four Operations with Negative Numbers 53 Addition and subtraction with negative numbers 54 Multiplication and division with negative numbers 56 Understanding Units 57 Adding and subtracting units 57 Multiplying and dividing units 57 Understanding Inequalities 58 Doesn’t equal (≠) 58 Less than (<) and greater than (>) 59 Less than or equal to (≤) and greater than or equal to (≥) 59 Approximately equals (≈) 60 Moving Beyond the Big Four: Exponents, Square Roots, and Absolute Value 60 Understanding exponents 60 Discovering your roots 61 Figuring out absolute value 62 Chapter 5: A Question of Values: Evaluating Arithmetic Expressions 63 Seeking Equality for All: Equations 64 Basic Math & Pre-Algebra For Dummies, 2nd Edition Hey, it’s just an expression 64 Evaluating the situation 65 Putting the Three E’s together 66 Introducing Order of Operations 66 Applying order of operations to Big Four expressions 67 Using order of operations in expressions with exponents 70 Understanding order of precedence in expressions with parentheses 71 Chapter 6: Say What? Turning Words into Numbers 75 Dispelling Two Myths about Word Problems 76 Word problems aren’t always hard 76 Word problems are useful 76 Solving Basic Word Problems 77 Turning word problems into word equations 77 Plugging in numbers for words 80 Solving More-Complex Word Problems 82 When numbers get serious 82 Too much information 83 Putting it all together 85 Chapter 7: Divisibility 87 Knowing the Divisibility Tricks 87 Counting everyone in: Numbers you can divide everything by 88 In the end: Looking at the final digits 88 Add it up: Checking divisibility by adding up digits 90 Ups and downs: Divisibility by 11 91 Identifying Prime and Composite Numbers 92 Chapter 8: Fabulous Factors and Marvelous Multiples 95 Knowing Six Ways to Say the Same Thing 96 Connecting Factors and Multiples 97 Finding Fabulous Factors 97 Deciding when one number is a factor of another 97 Understanding factor pairs 98 Generating a number’s factors 98 Identifying prime factors 100 Finding the greatest common factor (GCF) 105 Making Marvelous Multiples 105 Generating multiples 105 Finding the least common multiple (LCM) 106 Table of Contents Part 3: Parts of the Whole: Fractions, Decimals, and Percents 109 Chapter 9: Fooling with Fractions 111 Slicing a Cake into Fractions 112 Knowing the Fraction Facts of Life 114 Telling the numerator from the denominator 114 Flipping for reciprocals 114 Using ones and zeros 115 Mixing things up 115 Knowing proper from improper 115 Increasing and Reducing Terms of Fractions 116 Increasing the terms of fractions 117 Reducing fractions to lowest terms 118 Converting between Improper Fractions and Mixed Numbers 120 Knowing the parts of a mixed number 121 Converting a mixed number to an improper fraction 121 Converting an improper fraction to a mixed number 122 Understanding Cross-multiplication 122 Making Sense of Ratios and Proportions 123 Chapter 10: Parting Ways: Fractions and the Big Four Operations 125 Multiplying and Dividing Fractions 126 Multiplying numerators and denominators straight across 126 Doing a flip to divide fractions 127 All Together Now: Adding Fractions 129 Finding the sum of fractions with the same denominator 129 Adding fractions with different denominators 130 Taking It Away: Subtracting Fractions 137 Subtracting fractions with the same denominator 137 Subtracting fractions with different denominators 138 Working Properly with Mixed Numbers 140 Multiplying and dividing mixed numbers 141 Adding and subtracting mixed numbers 142 Chapter 11: Dallying with Decimals 149 Understanding Basic Decimal Stuff 150 Counting dollars and decimals 150 Identifying the place value of decimals 152 Knowing the decimal facts of life 153 Basic Math & Pre-Algebra For Dummies, 2nd Edition Performing the Big Four with Decimals 157 Adding decimals 158 Subtracting decimals 159 Multiplying decimals 160 Dividing decimals 161 Converting between Decimals and Fractions 165 Making simple conversions 165 Changing decimals to fractions 165 Changing fractions to decimals 167 Chapter 12: Playing with Percents 171 Making Sense of Percents 171 Dealing with Percents Greater than 100% 172 Converting to and from Percents, Decimals, and Fractions 173 Going from percents to decimals 173 Changing decimals into percents 174 Switching from percents to fractions 174 Turning fractions into percents 175 Solving Percent Problems 175 Figuring out simple percent problems 176 Turning the problem around 177 Deciphering more-difficult percent problems 178 Putting All the Percent Problems Together 178 Identifying the three types of percent problems 179 Solving percent problems with equations 180 Chapter 13: Word Problems with Fractions, Decimals, and Percents 183 Adding and Subtracting Parts of the Whole in Word Problems 184 Sharing a pizza: Fractions 184 Buying by the pound: Decimals 185 Splitting the vote: Percents 185 Problems about Multiplying Fractions 186 Renegade grocery shopping: Buying less than they tell you to 186 Easy as pie: Working out what’s left on your plate 187 Multiplying Decimals and Percents in Word Problems 188 To the end: Figuring out how much money is left 188 Finding out how much you started with 189 Handling Percent Increases and Decreases in Word Problems 191 Raking in the dough: Finding salary increases 191 Earning interest on top of interest 192 Getting a deal: Calculating discounts 193 Table of Contents Part 4: Picturing and Measuring — Graphs, Measures, Stats, and Sets 195 Chapter 14: A Perfect Ten: Condensing Numbers with Scientific Notation 197 First Things First: Using Powers of Ten as Exponents 198 Counting zeros and writing exponents 198 Adding exponents to multiply 200 Working with Scientific Notation 200 Writing in scientific notation 201 Seeing why scientific notation works 202 Understanding order of magnitude 203 Multiplying with scientific notation 203 Chapter 15: How Much Have You Got? Weights and Measures 205 Examining Differences between the English and Metric Systems 206 Looking at the English system 206 Looking at the metric system 208 Estimating and Converting between the English and Metric Systems 210 Estimating in the metric system 211 Converting units of measurement 213 Chapter 16: Picture This: Basic Geometry 217 Getting on the Plane: Points, Lines, Angles, and Shapes 218 Making some points 218 Knowing your lines 218 Figuring the angles 219 Shaping things up 221 Closed Encounters: Shaping Up Your Understanding of 2-D Shapes 221 Polygons 222 Circles 224 Taking a Trip to Another Dimension: Solid Geometry 225 The many faces of polyhedrons 225 3-D shapes with curves 226 Measuring Shapes: Perimeter, Area, Surface Area, and Volume 227 2-D: Measuring on the flat 228 Spacing out: Measuring in three dimensions 235 Basic Math & Pre-Algebra For Dummies, 2nd Edition Chapter 17: Seeing Is Believing: Graphing as a Visual Tool 239 Looking at Three Important Graph Styles 240 Bar graph 240 Pie chart 241 Line graph 242 Using the xy-Graph 243 Plotting points on an xy-graph 244 Drawing lines on an xy-graph 245 Chapter 18: Solving Geometry and Measurement Word Problems 247 The Chain Gang: Solving Measurement Problems with Conversion Chains 248 Setting up a short chain 248 Working with more links 249 Pulling equations out of the text 250 Rounding off: Going for the short answer 251 Solving Geometry Word Problems 253 Working from words and images 253 Breaking out those sketching skills 255 Chapter 19: Figuring Your Chances: Statistics and Probability 259 Gathering Data Mathematically: Basic Statistics 260 Understanding differences between qualitative and quantitative data 260 Working with qualitative data 261 Working with quantitative data 264 Looking at Likelihoods: Basic Probability 266 Figuring the probability 267 Oh, the possibilities! Counting outcomes with multiple coins 268 Chapter 20: Setting Things Up with Basic Set Theory 271 Understanding Sets 272 Elementary, my dear: Considering what’s inside sets 273 Sets of numbers 275 Performing Operations on Sets 275 Union: Combined elements 276 Intersection: Elements in common 276 Relative complement: Subtraction (sorta) 277 Complement: Feeling left out 277 Table of Contents Part 5: the X-files: Introduction to Algebra 279 Chapter 21: Enter Mr X: Algebra and Algebraic Expressions 281 Seeing How X Marks the Spot 282 Expressing Yourself with Algebraic Expressions 282 Evaluating algebraic expressions 283 Coming to algebraic terms 285 Making the commute: Rearranging your terms 286 Identifying the coefficient and variable 287 Identifying like terms 288 Considering algebraic terms and the Big Four 289 Simplifying Algebraic Expressions 292 Combining like terms 292 Removing parentheses from an algebraic expression 293 Chapter 22: Unmasking Mr X: Algebraic Equations 299 Understanding Algebraic Equations 300 Using X in Equations 300 Choosing among four ways to solve algebraic equations 301 The Balancing Act: Solving for x 302 Striking a balance 303 Using the Balance Scale to Isolate X 304 Rearranging Equations and Isolating x 305 Rearranging terms on one side of an equation 305 Moving terms to the other side of the equals sign 306 Removing parentheses from equations 307 Cross-multiplying 309 Chapter 23: Putting Mr X to Work: Algebra Word Problems 311 Solving Algebra Word Problems in Five Steps 312 Declaring a variable 312 Setting up the equation 313 Solving the equation 314 Answering the question 314 Checking your work 315 Choosing Your Variable Wisely 315 Solving More-Complex Algebraic Problems 316 Charting four people 316 Crossing the finish line with five people 318 Basic Math & Pre-Algebra For Dummies, 2nd Edition Part 6: the Part of Tens 321 Chapter 24: Ten Little Math Demons That Trip People Up 323 Knowing the Multiplication Table 324 Adding and Subtracting Negative Numbers 324 Multiplying and Dividing Negative Numbers 325 Knowing the Difference between Factors and Multiples 325 Reducing Fractions to Lowest Terms 326 Adding and Subtracting Fractions 326 Multiplying and Dividing Fractions 327 Identifying Algebra’s Main Goal: Find X 327 Knowing Algebra’s Main Rule: Keep the Equation in Balance 328 Seeing Algebra’s Main Strategy: Isolate x 328 Chapter 25: Ten Important Number Sets to Know 329 Counting on Counting (or Natural) Numbers 330 Identifying Integers 330 Knowing the Rationale behind Rational Numbers 331 Making Sense of Irrational Numbers 331 Absorbing Algebraic Numbers 332 Moving through Transcendental Numbers 333 Getting Grounded in Real Numbers 333 Trying to Imagine Imaginary Numbers 333 Grasping the Complexity of Complex Numbers 334 Going beyond the Infinite with Transfinite Numbers 335 Index 337

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Book SynopsisBob Blitzer is a native of Manhattan and received aBachelor of Arts degree with dual majors in mathematics and psychology (minor:English literature) from the City College of New York. His unusual combinationof academic interests led him toward a Master of Arts in mathematics from theUniversity of Miami and a doctorate in behavioral sciences from NovaUniversity. Bob's love for teaching mathematics was nourished for nearly 30years at Miami Dade College, where he received numerous teaching awards,including Innovator of the Year from the League for Innovations in theCommunity College and an endowed chair based on excellence in the classroom. Inaddition to Algebra and Trigonometry, Bob has written textbookscovering developmental mathematics, introductory algebra, intermediate algebra,trigonometry, precalculus, and liberal arts mathematics, all published byPearson. When not secluded in his Northern California writer's cabin, Bob canbe found hikinTable of Contents P. Prerequisites: Fundamental Concepts of Algebra P.1 Algebraic Expressions, Mathematical Models, and Real Numbers P.2 Exponents and Scientific Notation P.3 Radicals and Rational Exponents P.4 Polynomials P.5 Factoring Polynomials P.6 Rational Expressions 1. Equations and Inequalities 1.1 Graphs and Graphing Utilities 1.2 Linear Equations and Rational Equations 1.3 Models and Applications 1.4 Complex Numbers 1.5 Quadratic Equations 1.6 Other Types of Equations 1.7 Linear Inequalities and Absolute Value Inequalities 2. Functions and Graphs 2.1 Basics of Functions and Their Graphs 2.2 More on Functions and Their Graphs 2.3 Linear Functions and Slope 2.4 More on Slope 2.5Transformations of Functions 2.6 Combinations of Functions; Composite Functions 2.7 Inverse Functions 2.8 Distance and Midpoint Formulas; Circles 3. Polynomial and Rational Functions 3.1 Quadratic Functions 3.2 Polynomial Functions and Their Graphs 3.3 Dividing Polynomials; Remainder and Factor Theorems 3.4 Zeros of Polynomial Functions 3.5 Rational Functions and Their Graphs 3.6 Polynomial and Rational Inequalities 3.7 Modeling Using Variation 4. Exponential and Logarithmic Functions 4.1 Exponential Functions 4.2 Logarithmic Functions 4.3 Properties of Logarithms 4.4 Exponential and Logarithmic Equations 4.5 Exponential Growth and Decay; Modeling Data 5. Trigonometric Functions 5.1 Angles and Radian Measure 5.2 Right Triangle Trigonometry 5.3 Trigonometric Functions of Any Angle 5.4 Trigonometric Functions of Real Numbers; Periodic Functions 5.5 Graphs of Sine and Cosine Functions 5.6 Graphs of Other Trigonometric Functions 5.7 Inverse Trigonometric Functions 5.8 Applications of Trigonometric Functions 6. Analytic Trigonometry 6.1 Verifying Trigonometric Identities 6.2 Sum and Difference Formulas 6.3Double-Angle, Power-Reducing, and Half-Angle Formulas 6.4 Product-to-Sumand Sum-to-Product Formulas 6.5 Trigonometric Equations 7. Additional Topics in Trigonometry 7.1 The Law of Sines 7.2 The Law of Cosines 7.3 Polar Coordinates 7.4 Graphs of Polar Equations 7.5 Complex Numbers in Polar Form; De Moivre's Theorem 7.6 Vectors 7.7 The Dot Product 8. Systems of Equations and Inequalities 8.1 Systems of Linear Equations in Two Variables 8.2 Systems of Linear Equations in Three Variables 8.3 Partial Fractions 8.4 Systems of Nonlinear Equations in Two Variables 8.5 Systems of Inequalities 8.6 Linear Programming 9. Matrices and Determinants 9.1 Matrix Solutions to Linear Systems 9.2 Inconsistent and Dependent Systems and Their Applications 9.3 Matrix Operations and Their Applications 9.4 Multiplicative Inverses of Matrices and Matrix Equations 9.5 Determinants and Cramer's Rule 10. Conic Sections and Analytic Geometry 10.1 The Ellipse 10.2 The Hyperbola 10.3 The Parabola 10.4 Rotation of Axes 10.5 Parametric Equations 10.6 Conic Sections in Polar Coordinates 11. Sequences, Induction, and Probability 11.1 Sequences and Summation Notation 11.2 Arithmetic Sequences 11.3 Geometric Sequences and Series 11.4 Mathematical Induction 11.5 The Binomial Theorem 11.6 Counting Principles, Permutations, and Combinations 11.7 Probability

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Book SynopsisMarge Lial (late) was always interested in math; it was her favorite subject in the first grade! Marge's intense desire to educate both her students and herself has inspired the writing of numerous best-selling textbooks. Marge, who received Bachelor's and Master's degrees from California State University at Sacramento, was affiliated with American River College. An avid reader and traveler, her travel experiences often found their way into her books as applications, exercise sets, and feature sets. Her interest in archeology led to trips to various digs and ruin sites, producing some fascinating problems for her textbooks involving such topics as the building of Mayan pyramids and the acoustics of ancient ball courts in the Yucatan. When John Hornsby enrolled as an undergraduate at Louisiana State University, he was uncertain whether he wanted to study mathematics education or journalism. His ultimate decision was to become a teacher, but afTable of ContentsR. Review of Basic Concepts R.1. Fractions, Decimals, and Percents R.2. Sets and Real Numbers R.3. Real Number Operations and Properties R.4. Integer and Rational Exponents R.5. Polynomials R.6. Factoring Polynomials R.7. Rational Expressions R.8. Radical Expressions 1. Equations and Inequalities 1.1. Linear Equations 1.2. Applications and Modeling with Linear Equations 1.3. Complex Numbers 1.4. Quadratic Equations 1.5. Applications and Modeling with Quadratic Equations 1.6. Other Types of Equations and Applications 1.7. Inequalities 1.8. Absolute Value Equations and Inequalities 2. Graphs and Functions 2.1. Rectangular Coordinates and Graphs 2.2. Circles 2.3. Functions 2.4. Linear Functions 2.5. Equations of Lines and Linear Models 2.6. Graphs of Basic Functions 2.7. Graphing Techniques 2.8. Function Operations and Composition 3. Polynomial and Rational Functions 3.1. Quadratic Functions and Models 3.2. Synthetic Division 3.3. Zeros of Polynomial Functions 3.4. Polynomial Functions: Graphs, Applications, and Models 3.5. Rational Functions: Graphs, Applications, and Models 3.6. Polynomial and Rational Inequalities 3.7. Variation 4. Inverse, Exponential, and Logarithmic Functions 4.1. Inverse Functions 4.2. Exponential Functions 4.3. Logarithmic Functions 4.4. Evaluating Logarithms and the Change-of-Base Theorem 4.5. Exponential and Logarithmic Equations 4.6. Applications of Models of Exponential Growth and Decay 5. Trigonometric Functions 5.1. Angles 5.2. Trigonometric Functions 5.3. Trigonometric Function Values and Angle Measures 5.4. Solutions and Applications of Right Triangles 6. The Circular Functions and Their Graphs 6.1. Radian Measure 6.2. The Unit Circle and Circular Functions 6.3. Graphs of the Sine and Cosine Functions 6.4. Translations of the Graphs of the Sine and Cosine Functions 6.5. Graphs of the Tangent and Cotangent Functions 6.6. Graphs of the Secant and Cosecant Functions 6.7. Harmonic Motion 7. Trigonometric Identities and Equations 7.1. Fundamental Identities 7.2. Verifying Trigonometric Identities 7.3. Sum and Difference Identities 7.4. Double-Angle and Half-Angle Identities 7.5. Inverse Circular Functions 7.6. Trigonometric Equations 7.7. Equations Involving Inverse Trigonometric Functions 8. Applications of Trigonometry 8.1. The Law of Sines 8.2. The Law of Cosines 8.3. Geometrically Defined Vectors and Applications 8.4. Algebraically Defined Vectors and the Dot Product 8.5. Trigonometric (Polar) Form of Complex Numbers; Products and Quotients 8.6. De Moivre's Theorem; Powers and Roots of Complex Numbers 8.7. Polar Equations and Graphs 8.8. Parametric Equations, Graphs, and Applications 9. Systems and Matrices 9.1. Systems of Linear Equations 9.2. Matrix Solution of Linear Systems 9.3. Determinant Solution of Linear Systems 9.4. Partial Fractions 9.5. Nonlinear Systems of Equations 9.6. Systems of Inequalities and Linear Programming 9.7. Properties of Matrices 9.8. Matrix Inverses 10. Analytic Geometry 10.1. Parabolas 10.2. Ellipses 10.3. Hyperbolas 10.4. Summary of the Conic Sections 11. Further Topics in Algebra 11.1. Sequences and Series 11.2. Arithmetic Sequences and Series 11.3. Geometric Sequences and Series 11.4. The Binomial Theorem 11.5. Mathematical Induction 11.6. Basics of Probability Appendix A Polar Form of Conic Sections Appendix B Rotation of Axes Appendix C Geometry Formulas

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Book Synopsis

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Book Synopsis

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Book Synopsis

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